×

zbMATH — the first resource for mathematics

Jacobian-free Newton-Krylov methods: a survey of approaches and applications. (English) Zbl 1036.65045
Summary: Jacobian-free Newton-Krylov (JFNK) methods are synergistic combinations of Newton-type methods for superlinearly convergent solution of nonlinear equations and Krylov subspace methods for solving the Newton correction equations. The link between the two methods is the Jacobian-vector product, which may be probed approximately without forming and storing the elements of the true Jacobian, through a variety of means.
Various approximations to the Jacobian matrix may still be required for preconditioning the resulting Krylov iteration. As with Krylov methods for linear problems, successful application of the JFNK method to any given problem is dependent on adequate preconditioning. JFNK has potential for application throughout problems governed by nonlinear partial differential equations and integro-differential equations.
In this survey paper, we place JFNK in context with other nonlinear solution algorithms for both boundary value problems (BVPs) and initial value problems (IVPs). We provide an overview of the mechanics of JFNK and attempt to illustrate the wide variety of preconditioning options available. It is emphasized that JFNK can be wrapped (as an accelerator) around another nonlinear fixed point method (interpreted as a preconditioning process, potentially with significant code reuse).
The aim of this paper is not to trace fully the evolution of JFNK, nor to provide proofs of accuracy or optimal convergence for all of the constituent methods, but rather to present the reader with a perspective on how JFNK may be applicable to applications of interest and to provide sources of further practical information.

MSC:
65H10 Numerical computation of solutions to systems of equations
65F35 Numerical computation of matrix norms, conditioning, scaling
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65R20 Numerical methods for integral equations
35Q35 PDEs in connection with fluid mechanics
45G10 Other nonlinear integral equations
65Z05 Applications to the sciences
76M25 Other numerical methods (fluid mechanics) (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Allgower, E.; Georg, K., Continuation and path following, (), 1-65 · Zbl 0792.65034
[2] Anderson, D.A.; Tannehill, J.C.; Pletcher, R.H., Computational fluid dynamics and heat transfer, (1984), Hemisphere Publishing New York · Zbl 0569.76001
[3] Anderson, W.K.; Bonhaus, D.L., An implicit upwind algorithm for computing turbulent flows on unstructured grids, Comp. fluids, 23, 1-21, (1994) · Zbl 0806.76053
[4] Anderson, W.K.; Rausch, R.D.; Bonhaus, D.L., Implicit/multigrid algorithms for incompressible turbulent flows on unstructured grids, J. comput. phys., 128, 391-408, (1996) · Zbl 0862.76045
[5] Ashby, S.F.; Falgout, R.D., A parallel multigrid preconditioned conjugate gradient algorithm for groundwater flow simulations, Nucl. sci. engrg., 124, 145-159, (1996)
[6] Axelsson, O., Iterative solution methods, (1994), Cambridge University Press Cambridge · Zbl 0795.65014
[7] S. Balay, W.D. Gropp, L.C. McInnes, B.F. Smith, PETSc users manual, Technical Report, ANL-95/11 - Revision 2.1.0, Argonne National Laboratory, April 2001
[8] Balsara, D., Fast and accurate discrete ordinates methods for multidimensional radiative transfer. part I, basic methods, J. quant. spectr. radiat. transfer, 69, 671-707, (2001)
[9] Banas, K., A newton – krylov solver with multiplicative Schwarz preconditioning for finite element compressible flow simulations, Comm. numer. methods engrg., 18, 275-296, (2002) · Zbl 1009.76048
[10] Bank, R.; Chan, T.; Coughran, W.; Smith, R., The alternate block factorization procedure for systems of partial differential equations, Bit, 29, 938-954, (1989) · Zbl 0715.65097
[11] Barrett, R.; Berry, M.; Chan, T.F.; Demmel, J.; Donato, J.; Dongarra, J.; Eijkhout, V.; Pozo, R.; Romine, C.; van der Vorst, H., Templates for the solution of linear systems: building blocks for iterative methods, SIAM, (1994)
[12] Bates, J.W.; Knoll, D.A.; Rider, W.J.; Lowrie, R.B.; Mousseau, V.A., On consistent time-integration methods for radiation hydrodynamics in the equilibrium diffusion limit: low energy density regime, J. comput. phys., 167, 99-130, (2001) · Zbl 1052.76041
[13] Benzi, M., Preconditioning techniques for large linear systems: a survey, J. comput. phys., 182, 418-477, (2002) · Zbl 1015.65018
[14] Biros, G.; Ghattas, O., Parallel newton – krylov methods for PDE-constrained optimization, () · Zbl 1091.65061
[15] Biros, G.; Ghattas, O., A Lagrange-newton – krylov – schur method for PDE-constrained optimization, SIAG/OPT views-and-news, 11, 1-6, (2000)
[16] Bischof, C.; Carle, A.; Corliss, G.; Griewank, A.; Hovland, P., ADIFOR - generating derivative codes from Fortran programs, Sci. program., 1, 1-29, (1992)
[17] Bischof, C.; Roh, L.; Mauer, A., ADIC - an extensible automatic differentiation tool for ANSI-C, Software – practice exp., 27, 1427-1456, (1997)
[18] Bjørstad, P., Fast numerical solution of the biharmonic Dirichlet problem on rectangles, SIAM J. numer. anal., 20, 59-71, (1983) · Zbl 0561.65077
[19] Bosisio, F.; Micheletti, S.; Sacco, R., A discretization scheme for an extended drift-diffusion model including trap-assisted phenomena, J. comput. phys., 159, 197-212, (2000) · Zbl 1073.82624
[20] Bowers, R.L.; Wilson, J.R., Numerical modeling in applied physics and astrophysics, (1991), Jones and Bartlett Boston · Zbl 0786.76001
[21] Brackbill, J.U.; Knoll, D.A., Transient magnetic reconnection and unstable shear layers, Phys. rev. lett., 86, 2329-2332, (2001)
[22] Brandt, A., Multi-level adaptive solutions to boundary value problems, Math. comp., 31, 333, (1977) · Zbl 0373.65054
[23] A. Brandt, Multigrid techniques: 1984 guide with applications to fluid dynamics, Technical Report, von Karman Institute, 1984 · Zbl 0581.76033
[24] Brown, P.N., A local convergence theory for combined inexact-Newton/finite-difference projection methods, SIAM J. numer. anal., 24, 407-434, (1987) · Zbl 0618.65037
[25] Brown, P.N.; Hindmarsh, A.C., Matrix-free methods for stiff systems of ode’s, SIAM J. numer. anal., 23, 610-638, (1986) · Zbl 0615.65078
[26] Brown, P.N.; Hindmarsh, A.C., Reduced storage matrix methods in stiff ODE systems, J. appl. math. comput., 31, 40-91, (1989) · Zbl 0677.65074
[27] Brown, P.N.; Saad, Y., Hybrid Krylov methods for nonlinear systems of equations, SIAM J. sci. stat. comput., 11, 450-481, (1990) · Zbl 0708.65049
[28] Brown, P.N.; Saad, Y., Convergence theory for nonlinear newton – krylov algorithms, SIAM J. opt., 4, 297-330, (1994) · Zbl 0814.65048
[29] Brown, P.N.; Woodward, C., Preconditioning strategies for fully implicit radiation diffusion with material-energy coupling, SIAM J. sci. comput., 23, 499-516, (2001) · Zbl 0992.65102
[30] X.-C. Cai, M. Dryja, M. Sarkis, RASHO: a restricted additive Schwarz preconditioner with harmonic overlap, in: Proceedings of the 13th International Conference on Domain Decomposition Methods, CIMNE, 2002, pp. 337-344 · Zbl 1026.65034
[31] Cai, X.-C.; Farhat, C.; Sarkis, M., Schwarz methods for the unsteady compressible navier – stokes equations on unstructured meshes, (), 453-460
[32] X.-C. Cai, C. Farhat, M. Sarkis, A minimum overlap restricted additive Schwarz preconditioner and applications in 3d flow simulations, in: Proceedings of the Tenth International Conference on Domain Decomposition Methods, AMS, 1998, pp. 238-244 · Zbl 0936.76036
[33] Cai, X.-C.; Gropp, W.D.; Keyes, D.E.; Melvin, R.G.; Young, D.P., Parallel newton – krylov – schwarz algorithms for the transonic full potential equation, SIAM J. sci. comput., 19, 246-265, (1998) · Zbl 0917.76035
[34] Cai, X.-C.; Gropp, W.D.; Keyes, D.E.; Tidriri, M.D., Newton – krylov – schwarz methods in CFD, (), 17-30 · Zbl 0876.76059
[35] Cai, X.-C.; Keyes, D.E., Nonlinearly preconditioned inexact Newton algorithms, SIAM J. sci. comput., 24, 183-200, (2002) · Zbl 1015.65058
[36] Cai, X.-C.; Keyes, D.E.; Marcinkowski, L., Nonlinear additive Schwarz preconditioners and applications in computational fluid dynamics, Int. J. numer. methods fluids, 40, 1463-1470, (2002) · Zbl 1025.76040
[37] Cai, X.-C.; Keyes, D.E.; Venkatakrishnan, V., Newton – krylov – schwarz: an implicit solver for CFD, (), 387-400
[38] Cai, X.-C.; Keyes, D.E.; Young, D.P., A nonlinearly additive Schwarz preconditioned inexact Newton method for shocked duct flow, (), 345-352 · Zbl 1201.76107
[39] Cai, X.-C.; Sarkis, M., A restricted additive Schwarz preconditioner for general sparse linear systems, SIAM J. sci. stat. comput., 21, 792-797, (1999) · Zbl 0944.65031
[40] L. Chacon, Fokker-Planck modeling of spherical inertial electrostatic, virtual-cathode fusion systems, PhD thesis, University of Illinois, 2000
[41] Chacon, L.; Barnes, D.C.; Knoll, D.A.; Miley, G.H., An implicit energy-conservative 2D fokker – planck algorithm: II Jacobian-free newton – krylov solver, J. comput. phys., 157, 654-682, (2000) · Zbl 0961.76058
[42] Chacon, L.; Barnes, D.C.; Knoll, D.A.; Miley, G.H., A bounce-averaged fokker – planck code for Penning fusion devices, Comput. phys. comm., 134, 182-208, (2001) · Zbl 1056.76061
[43] Chacon, L.; Knoll, D.A., A 2D high-β Hall MHD implicit nonlinear solver, J. comput. phys., 188, 573-592, (2003) · Zbl 1127.76375
[44] Chacon, L.; Knoll, D.A.; Finn, J.M., An implicit nonlinear reduced resistive MHD solver, J. comput. phys., 178, 15-36, (2002) · Zbl 1139.76328
[45] Chacon, L.; Knoll, D.A.; Finn, J.M., Hall MHD effects in the 2-d kelvin – helmholtz/tearing instability, Phys. lett. A, 308, 187-197, (2003) · Zbl 1086.81559
[46] Chacon, L.; Miley, G.H.; Barnes, D.C.; Knoll, D.A., Energy gain calculations in Penning fusion systems using a bounce-averaged fokker – planck model, Phys. plasmas, 7, 4547-4560, (2000)
[47] Chan, T.F.; Jackson, K.R., Nonlinearly preconditioned Krylov subspace methods for discrete Newton algorithms, SIAM J. sci. stat. comput., 5, 533-542, (1984) · Zbl 0574.65043
[48] T.S. Coffey, C.T. Kelley, D.E. Keyes, Pseudo-transient continuation and differential-algebraic equations. SIAM J. Sci. Comput., 2003 (to appear) · Zbl 1048.65080
[49] Culler, D.E.; Singh, J.P.; Gupta, A., Parallel computer architecture, (1998), Morgan-Kaufmann Los Altos, CA
[50] Cummins, S.J.; Brackbill, J.U., An implicit particle-in-cell method for granular materials, J. comput. phys., 180, 506-548, (2002) · Zbl 1143.74388
[51] Davis, G.D.V., Natural convection of air in a square cavity: a benchmark numerical solution, Int. J. numer. methods fluids, 3, 249, (1983)
[52] Dawson, C.; Klie, H.; Wheeler, M.; Woodward, C., A parallel, implicit, cell centered method for two-phase flow with a preconditioned newton – krylov solver, Comp. geosci., 1, 215-249, (1997) · Zbl 0941.76062
[53] Dembo, R., Inexact Newton methods, SIAM J. numer. anal., 19, 400-408, (1982) · Zbl 0478.65030
[54] Dennis, J.E.; Schnabel, R.B., Numerical methods for unconstrained optimization and nonlinear equa tions, (1983), Prentice-Hall Englewood Cliffs, NJ
[55] M. Dryja, O.B. Widlund, An additive variant of the Schwarz alternating method for the case of many subregions, Technical Report 339, Courant Institute, NYU, 1987
[56] Eisenstat, S.C.; Walker, H.F., Globally convergent inexact Newton methods, SIAM J. optim., 4, 393-422, (1994) · Zbl 0814.65049
[57] Eisenstat, S.C.; Walker, H.F., Choosing the forcing terms in a inexact Newton method, SIAM J. sci. comput., 17, 16-32, (1996) · Zbl 0845.65021
[58] Epperlein, E.M., Implicit and conservative difference scheme for the fokker – planck equation, J. comput. phys., 112, 291, (1994) · Zbl 0806.76050
[59] Ern, A.; Giovangigli, V.; Keyes, D.E.; Smooke, M.D., Towards polyalgorithmic linear system solvers for nonlinear elliptic systems, SIAM J. sci. comput., 15, 681-703, (1994) · Zbl 0807.65027
[60] Freund, R., A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems, SIAM J. sci. comput., 14, 470-482, (1993) · Zbl 0781.65022
[61] M. Garbey, R. Hoppe, D.E. Keyes, Y. Kuznetsov, J. Périaux (Eds.), Proceedings of the 13th International Conference on Domain Decomposition Methods, CIMNE, 2002. Available from
[62] Gear, C.W.; Saad, Y., Iterative solution of linear equations in ODE codes, SIAM J. sci. stat. comput., 4, 583-601, (1983) · Zbl 0541.65051
[63] Geuzaine, P., Newton – krylov strategy for compressible turbulent flows on unstructured mesh, Aiaa j., 39, 528-531, (2001)
[64] Ghia, U.; Ghia, K.; Shin, C., High-re solutions for incompressible flow using the navier – stokes equations and a multigrid method, J. comput. phys., 48, 387, (1982) · Zbl 0511.76031
[65] Glowinski, R., Numerical methods for nonlinear variational problems, (1984), Springer Verlag Berlin · Zbl 0575.65123
[66] Golub, G.H.; O’Leary, D., Some history of the conjugate gradient and Lanczos algorithms: 1948-1976, SIAM rev., 50-102, (1989) · Zbl 0673.65017
[67] Greenbaum, A., Iterative methods for solving linear systems, (1997), SIAM Philadelphia, PA · Zbl 0883.65022
[68] Gropp, W.; Keyes, D.E.; McInnes, L.; Tidriri, M., Globalized newton – krylov – schwarz algorithms and software for parallel implicit CFD, Int. J. high performance comput. appl., 14, 102-136, (2000)
[69] Gropp, W.D.; Kaushik, D.K.; Keyes, D.E.; Smith, B.F., High performance parallel implicit CFD, Parallel comput., 27, 337-362, (2001) · Zbl 0971.68191
[70] Gummel, H., A self-consistent iterative scheme for one-dimensional steady state transistor calculations, IEEE trans. electron. dev., ED-11, 455-465, (1964)
[71] Gutknecht, M.H., Lanczos-type solvers for nonsymmetric linear systems of equations, (), 271-398 · Zbl 0888.65030
[72] Hackbusch, W., Iterative methods for large sparse linear systems, (1993), Springer Berlin
[73] Hammond, G.E.; Valocchi, A.J.; Lichtner, P.C., Modeling multicomponent reactive transport on parallel computers using Jacobian-free newton – krylov with operator-splt preconditioning, Devel. water resour., 47, 727-734, (2002)
[74] Harlow, F.; Amsden, A., A numerical fluid dynamical calculation method for all flow speeds, J. comput. phys., 8, 197-214, (1971) · Zbl 0221.76011
[75] Hestenes, M.R.; Stiefel, E.L., Methods of conjugate gradients for solving linear systems, J. res. natl. bureau of standards, section B, 49, 409-436, (1952) · Zbl 0048.09901
[76] Hockney, R., Meth. comp. phys., 9, 135-211, (1970)
[77] Hovland, P.; McInnes, L.C., Parallel simulation of compressible flow using automatic differentiation and petsc, Parallel comput., 27, 503-519, (2001) · Zbl 0972.68165
[78] Hutchinson, B.R.; Raithby, G.D., A multigrid method based on the additive correction strategy, Numer. heat transfer, 9, 511-537, (1986)
[79] Jenkins, E.W.; Kees, C.E.; Kelley, C.T.; Miller, C.T., An aggregation-based domain decomposition preconditioner for groundwater flow, SIAM J. sci. stat. comput., 23, 430-441, (2001) · Zbl 1036.65109
[80] Jiang, H.; Forsyth, P.A., Robust linear and nonlinear strategies for solution of the transonic Euler equations, Comp. fluids, 24, 753-770, (1995) · Zbl 0845.76070
[81] Johnson, R.W.; McHugh, P.R.; Knoll, D.A., High-order scheme implementation using newton – krylov solution methods, Numer. heat trans., part B, 31, 295-312, (1997)
[82] Jones, J.E.; Woodward, C.S., Newton – krylov-multigrid solvers for large scale, highly heterogeneous, variably saturated flow problems, Adv. water resour., 24, 763-774, (2001)
[83] Kaushik, D.E.; Keyes, D.E.; Smith, B.F., On the interaction of architecture and algorithm in the domain-based parallelization of an unstructured grid incompressible flow code, (), 311-319
[84] Kelley, C.T., Iterative methods for linear and nonlinear equations, (1995), SIAM Philadelphia · Zbl 0832.65046
[85] Kelley, C.T.; Keyes, D.E., Convergence analysis of pseudo-transient continuation, SIAM J. numer. anal., 35, 508-523, (1998) · Zbl 0911.65080
[86] Kerkhoven, T.; Saad, Y., On acceleration methods for coupled nonlinear elliptic systems, Numer. math., 60, 525-548, (1992) · Zbl 0724.65095
[87] Keyes, D.E., Domain decomposition methods for the parallel computation of reacting flows, Comp. phys. commun., 53, 181-200, (1989) · Zbl 0804.76066
[88] Keyes, D.E., Aerodynamic applications of newton – krylov – schwarz solvers, (), 1-20 · Zbl 0854.76073
[89] Keyes, D.E., How scalable is domain decomposition in practice?, (), 286-297
[90] Keyes, D.E.; Gropp, W.D., A comparison of domain decomposition techniques for elliptic partial differential equations and their parallel implementation, SIAM J. sci. stat. comput., 8, s166-s202, (1987) · Zbl 0619.65088
[91] Keyes, D.E.; Kaushik, D.K.; Smith, B.F., Prospects for CFD on petaflops systems, (), 1079-1096
[92] Keyes, D.E.; McInnes, L.; Samyono, W., Using automatic differentiation for second-order matrix-free methods in PDE-constrained optimization, (), 35-50
[93] Keyes, D.E.; Venkatakrishnan, V., Newton – krylov – schwarz methods: interfacing sparse linear solvers with nonlinear applications, Z. angew. math. mech., 76, Suppl. 1, 147-150, (1996) · Zbl 0900.65158
[94] D.A. Knoll, Development and application of a direct Newton solver for the two-dimensional tokamak edge plasma fluid equations, PhD thesis, University of New Mexico, 1991
[95] Knoll, D.A., An improved convection scheme applied to recombining divertor plasma flows, J. comput. phys., 142, 473-488, (1998) · Zbl 0932.76044
[96] Knoll, D.A.; Brackbill, J.U., The kelvin – helmholtz instability, differential rotation, and 3-D, localized, magnetic reconnection, Phys. plasmas, 9, 3775-3782, (2002)
[97] Knoll, D.A.; Chacon, L., Magnetic reconnection in the two-dimensional kelvin – helmholtz instability, Phys. rev. lett., 88, (2002), p. art. no. 215003
[98] Knoll, D.A.; Chacon, L.; Margolin, L.G.; Mousseau, V.A., On balanced approximations for the time integration of multiple time scale systems, J. comput. phys., 185, 583-611, (2003) · Zbl 1047.76074
[99] Knoll, D.A.; Kothe, D.B.; Lally, B., A new nonlinear solution method for phase-change problems, Numer. heat trans., part B, 35, 439-459, (1999)
[100] Knoll, D.A.; Lapenta, G.; Brackbill, J., A multilevel iterative field solver for implict, kinetic plasma simulation, J. comput. phys., 149, 377-388, (1999) · Zbl 0934.76048
[101] Knoll, D.A.; McHugh, P.R., NEWEDGE: a 2-D fully implicit edge plasma fluid code for advanced physics and complex geometries, J. nucl. mater., 196-198, 352-356, (1992)
[102] Knoll, D.A.; McHugh, P.R., An inexact Newton algorithm for solving the tokamak edge plasma fluid equations on a multiply connected domain, J. comput. phys., 116, 281-291, (1995) · Zbl 0818.76069
[103] Knoll, D.A.; McHugh, P.R., Newton – krylov methods applied to a system of convection – diffusion – reaction equations, Comput. phys. commun., 88, 141-160, (1995) · Zbl 0923.76199
[104] Knoll, D.A.; McHugh, P.R., Enhanced nonlinear iterative techniques applied to a nonequilibrium plasma flow, SIAM J. sci. comput., 19, 291-301, (1998) · Zbl 0913.76067
[105] Knoll, D.A.; McHugh, P.R.; Keyes, D.E., Newton – krylov methods for low Mach number compressible combustion, Aiaa j., 34, 961-967, (1996) · Zbl 0900.76406
[106] Knoll, D.A.; McHugh, P.R.; Krasheninnikov, S.I.; Sigmar, D.J., Simulation of dense recombining divertor plasmas with a navier – stokes neutral transport model, Phys. plasmas, 3, 293-303, (1996)
[107] Knoll, D.A.; McHugh, P.R.; Mousseau, V.A., Newton – krylov – schwarz methods applied to the tokamak edge plasma fluid equations, (), 75-96 · Zbl 0842.76061
[108] Knoll, D.A.; Mousseau, V.A., On newton – krylov multigrid methods for the incompressible navier – stokes equations, J. comput. phys., 163, 262-267, (2000) · Zbl 0994.76055
[109] Knoll, D.A.; Prinja, A.K.; Campbell, R.B., A direct Newton solver for the two-dimensional tokamak edge plasma fluid equations, J. comput. phys., 104, 418-426, (1993) · Zbl 0800.76339
[110] Knoll, D.A.; Rider, W.J., A multigrid preconditioned newton – krylov method, SIAM J. sci. comput., 21, 691-710, (2000) · Zbl 0952.65102
[111] Knoll, D.A.; Rider, W.J.; Olson, G.L., An efficient nonlinear solution method for non-equilibrium radiation diffusion, J. quant. spectrosc. radiat. transfer, 63, 15-29, (1999)
[112] Knoll, D.A.; Rider, W.J.; Olson, G.L., Nonlinear convergence, accuracy, and time step control in non-equilibrium radiation diffusion, J. quant. spectrosc. radiat. transfer, 70, 25-36, (2001)
[113] Knoll, D.A.; VanderHeyden, W.B.; Mousseau, V.A.; Kothe, D.B., On preconditioning newton – krylov methods in solidifying flow applications, SIAM J. sci. comput., 23, 381-397, (2002) · Zbl 1125.65315
[114] Krasheninnikov, S.I., Plasma recombination in tokamak divertors and divertor simulators, Phys. plasmas, 4, 1638, (1997)
[115] Larzelere, A.R., Creating simulation capabilities, IEEE comput. sci. engrg., 5, 27-35, (1988)
[116] MacArthur, J.W.; Patankar, S.V., Robust semidirect finite difference methods for solving the navier – stokes and energy equations, Int. J. numer. methods fluids, 9, 325-340, (1989)
[117] Mavriplis, D.J., Multigrid strategies for viscous flow solvers on anisotropic unstructured meshes, J. comput. phys., 145, 141, (1998) · Zbl 0926.76066
[118] Mavriplis, D.J., An assessment of linear versus non-linear multigrid methods for unstructured mesh solvers, J. comput. phys., 175, 302-325, (2002) · Zbl 0995.65099
[119] Mavriplis, D.J.; Venkatakrishnan, V., Agglomeration multigrid for the two-dimensional viscous flows, Comp. fluids, 24, 533, (1995) · Zbl 0846.76047
[120] McCormick, S.F., Multilevel adaptive methods for partial differential equations, (1989), SIAM Philadelphia · Zbl 0707.65080
[121] McHugh, P.R.; Knoll, D.A., Fully implicit finite volume solutions of the incompressible navier – stokes and energy equations using inexact newton’s method, Int. J. numer. methods fluids, 18, 439-455, (1994) · Zbl 0814.76071
[122] McHugh, P.R.; Knoll, D.A., Inexact newton’s method solution to the incompressible navier – stokes and energy equations using standard and matrix-free implementations, Aiaa j., 32, (1994)
[123] McHugh, P.R.; Knoll, D.A.; Keyes, D.E., Application of a newton – krylov – schwarz algorithm to low Mach number combustion, Aiaa j., 36, 290-292, (1998) · Zbl 0900.76422
[124] P.R. McHugh, D.A. Knoll, V.A. Mousseau, G.A. Hansen, An investigation of Newton-Krylov solution techniques for low mach number compressible flow, in: Proceedings of the ASME Fluids Engineering Division Summer Meeting, 1995
[125] Meza, J.C.; Tuminaro, R.S., A multigrid preconditioner for the semiconductor equations, SIAM J. sci. comput., 17, 118-132, (1996) · Zbl 0845.35122
[126] Mousseau, V.; Knoll, D.A.; Reisner, J., An implicit nonlinearly consistent method for the two-dimensional shallow-water equations with Coriolis force, Mon. weather rev., 130, 2611-2625, (2002)
[127] Mousseau, V.A.; Knoll, D.A., Fully implicit kinetic solution of collisional plasmas, J. comput. phys., 136, 308-323, (1997) · Zbl 0896.76057
[128] Mousseau, V.A.; Knoll, D.A., New physics-based preconditioning of implicit methods for non-equilibrium radiation diffusion, J. comput. phys., (2003) · Zbl 1027.65129
[129] Mousseau, V.A.; Knoll, D.A.; Rider, W.J., A multigrid newton – krylov solver for nonlinear systems, (), 200-206 · Zbl 0973.65041
[130] Mousseau, V.A.; Knoll, D.A.; Rider, W.J., Physics-based preconditioning and the newton – krylov method for non-equilibrium radiation diffusion, J. comput. phys., 160, 743-765, (2000) · Zbl 0949.65092
[131] Mukadi, L.S.; Hayes, R.E., Modelling the three-way catalytic converter with mechanistic kinetics using the newton – krylov method on a parallel computer, Comput. chem. engrg., 26, 439-455, (2002)
[132] Mulder, W.; Leer, B.V., Experiments with implicit upwind methods for the Euler equations, J. comput. phys., 59, 232-246, (1985) · Zbl 0584.76014
[133] Nachtigal, N.M.; Reddy, S.C.; Trefethen, L.N., How fast are nonsymmetric matrix iterations?, SIAM J. matrix anal. appl., 13, 778-795, (1992) · Zbl 0754.65036
[134] Nemec, M.; Zingg, D.W., Newton – krylov algorithm for aerodynamic design using the navier – stokes equations, Aiaa j., 40, 1146-1154, (2002)
[135] Oosterlee, C.W.; Washio, T., An evaluation of parallel multigrid as a solver and a preconditioner for singularly perturbed problems, SIAM J. sci. stat. comput., 19, 87-110, (1998) · Zbl 0913.65109
[136] Oran, E.S.; Boris, J.P., Numerical simulation of reactive flow, (1987), Elsevier New York · Zbl 0762.76098
[137] Ortega, J.; Rheinboldt, W., Iterative solution of nonlinear equations in several variables, (1970), Academic Press Boston · Zbl 0241.65046
[138] Patankar, S.V., Numerical heat transfer and fluid flow, (1980), Hemisphere Publishing New York · Zbl 0595.76001
[139] Pernice, M., A hybrid multigrid method for the steady-state incompressible navier – stokes equations, Elec. trans. numer. anal., 10, 74-91, (2000) · Zbl 0963.76064
[140] M. Pernice, Private communication, 2003
[141] Pernice, M.; Tocci, M., A multigrid-preconditioned newton – krylov method for the incompressible navier – stokes equations, SIAM J. sci. comput., 23, 398-418, (2001) · Zbl 0995.76061
[142] Pernice, M.; Walker, H.F., NITSOL: a Newton iterative solver for nonlinear systems, SIAM J. sci. comput., 19, 302-318, (1998) · Zbl 0916.65049
[143] M. Pernice, L. Zhou, H.F. Walker, Parallel solution of nonlinear partial differential equations using a globalized inexact Newton-Krylov-Schwarz method, Technical Report 48, University of Utah Center for High Performance Computing, 1997
[144] Porter, G.D., Detailed comparison of simulated and measured plasma profiles in the scrape-off layer and edge plasma of DIII-D, Phys. plasmas, 7, 3663-3680, (2000)
[145] Qin, N.; Ludlow, D.K.; Shaw, S.T., A matrix-free preconditioned Newton/GMRES method for unsteady navier – stokes solutions, Int. J. numer. methods fluids, 33, 223-248, (2000) · Zbl 0976.76049
[146] Freund, G.H.G.R.W.; Nachtigal, N.M., Iterative solution of linear systems, (), 57-100 · Zbl 0762.65019
[147] Rasetarinera, P.; Hussaini, M., An efficient implicit discontinuous spectral Galerkin method, J. comput. phys., 172, 718-738, (2001) · Zbl 0986.65093
[148] Reid, J.K., On the method of conjugate gradients for the solution of large sparse systems of linear equations, (), 231-254
[149] Reisner, J.; Mousseau, V.; Knoll, D.A., Application of the newton – krylov method to geophysical flows, Mon. weather rev., 129, 2404-2415, (2001)
[150] Reisner, J.; Wynne, S.; Margolin, L.; Linn, R., Coupled atmospheric-fire modeling employing the method of averages, Mon. weather rev., 128, 3683-3691, (2000)
[151] Reisner, J.; Wyszogrodzki, A.; Mousseau, V.; Knoll, D.A., An efficient physics-based preconditioner for the fully implicit solution of small-scale thermally driven atmospheric flows, J. comput. phys., 189, 30-44, (2003) · Zbl 1097.76543
[152] Rider, W.J.; Knoll, D.A., Time step size selection for radiation diffusion calculations, J. comput. phys., 152, 790-795, (1999) · Zbl 0940.65101
[153] Rider, W.J.; Knoll, D.A.; Olson, G.L., A multigrid preconditioned newton – krylov method for multimaterial equilibrium radiation diffusion, J. comput. phys., 152, 164-191, (1999) · Zbl 0944.85002
[154] Rognlien, T.D.; Milovich, J.L.; Rensink, M.E.; Porter, G.D., A fully implicit, time-dependent 2-d fluid code for modeling tokamak edge plasmas, J. nucl. mater., 196-198, 347-351, (1992)
[155] Rognlien, T.D.; Xu, X.Q.; Hindmarsh, A.C., Application of parallel implicit methods to edge-plasma numerical simulations, J. comput. phys, 175, 249-268, (2002) · Zbl 1168.76353
[156] Saad, Y., A flexible inner – outer preconditioned GMRES algorithm, SIAM J. sci. stat. comput., 14, 461-469, (1993) · Zbl 0780.65022
[157] Saad, Y., Iterative methods for sparse linear systems, (1996), PWS Publishing Company Boston · Zbl 1002.65042
[158] Saad, Y.; Schultz, M.H., GMRES: a generalized minimal residual algorithm for solving non-symetric linear systems, SIAM J. sci. stat. comput., 7, 856, (1986) · Zbl 0599.65018
[159] Saad, Y.; van der Vorst, H.A., Iterative solution of linear systems in the 20th century, J. comp. appl. math., 123, 1-33, (2000) · Zbl 0965.65051
[160] Schaffer, S., A semi-coarsening multigrid method for elliptic partial differential equations with highly discontinuous and anisotropic coefficients, SIAM J. sci. stat. comput., 20, 228-242, (1999) · Zbl 0913.65111
[161] Settari, A.; Aziz, K., A generalization of the additive correction methods for the iterative solution of matrix equations, SIAM J. numer. anal., 10, 506-521, (1973) · Zbl 0256.65020
[162] Shadid, J.N., Efficient parallel computation of unstructured finite element reacting flow solutions, Parallel comput., 23, 1307-1325, (1997) · Zbl 0894.68019
[163] Shadid, J.N.; Tuminaro, R.S., A comparison of preconditioned nonsymmetric Krylov methods on a large-scale mimd machine, SIAM J. sci. comput., 15, 440-459, (1994) · Zbl 0798.65047
[164] Shadid, J.N.; Tuminaro, R.S.; Walker, H.F., An inexact Newton method for fully coupled solution of the navier – stokes equations with heat and mass transport, J. comput. phys., 137, 155-185, (1997) · Zbl 0898.76066
[165] Smooke, M.D., Solution of burner-stabilized premixed laminar flames by boundary value methods, J. comput. phys., 48, 72-105, (1982) · Zbl 0492.65065
[166] Smooke, M.D.; Mattheij, R.M., On the solution of nonlinear two-point boundary value problems on successively refined grids, Appl. numer. math., 1, 463-487, (1985) · Zbl 0619.65074
[167] Smooke, M.D.; Mitchell, R.E.; Keyes, D.E., Numerical solution of two-dimensional axisymmetric laminar diffusion flames, Combust. sci. technol., 67, 85-122, (1989) · Zbl 0696.65087
[168] Suetomi, E., Two-dimensional fluid simulation of plasma reactors for the immobilization of krypton, Comput. phys. commun., 125, 60-74, (2000) · Zbl 0978.76064
[169] M.D. Tidriri, Schwarz-based algorithms for compressible flows, Technical Report 96-4, ICASE, January 1996
[170] Tidriri, M.D., Preconditioning techniques for the newton – krylov solution of compressible flows, J. comput. phys., 132, 51-61, (1997) · Zbl 0879.76063
[171] M.D. Tidriri, Hybrid Newton-Krylov/domain decomposition methods for compressible flows, in: Proceedings of the Ninth International Conference on Domain Decomposition Methods in Sciences and Engineering, 1998, pp. 532-539
[172] Tidriri, M.D., Development and study of newton – krylov – schwarz algorithms, Int. J. comput. fluid dyn., 15, 115-126, (2001) · Zbl 1017.76055
[173] Tocci, M.D.; Kelley, C.T.; Miller, C.T.; Kees, C.E., Inexact Newton methods and the method of lines for solving richards’ equation in two space dimensions, Comp. geosci., 2, 291-309, (1998) · Zbl 0962.76590
[174] Trottenberg, U.; Schuller, A.; Oosterlee, C., Multigrid, (2000), Academic Press New York
[175] R.S. Tuminaro, M. Heroux, S.A. Hutchinson, J.N. Shadid, Official Aztec user’s guide, Technical Report SAND99-8801J, Sandia National Laboratory, December 1999
[176] Tuminaro, R.S.; Walker, H.F.; Shadid, J.N., On backtracking failure in Newton-GMRES methods with a demonstration for the navier – stokes equations, J. comput. phys., 180, 549-558, (2002) · Zbl 1143.76489
[177] Turner, K.; Walker, H.F., Efficient high-accuracy solutions with GMRES(m), SIAM J. sci. stat. comput., 13, 815-825, (1992) · Zbl 0758.65029
[178] US Department of Energy, Scientific discovery through advanced computing, Technical Report, US Department of Energy, March 2000
[179] van der Vorst, H.A., Bi-CGSTAB: a fast and smoothly converging variant of bi-CG for the solution of nonsymmetric linear systems, SIAM J. sci. stat. comput., 13, 631-644, (1992) · Zbl 0761.65023
[180] van der Vorst, H.A.; Vuik, C., A comparison of some GMRES-like methods, Linear algebra appl., 160, 131-162, (1992) · Zbl 0749.65027
[181] S.P. Vanka, G.K. Leaf, Fully coupled solution of pressure-linked fluid-flow equations, Technical Report ANL-83-73, Argonne National Laboratory, 1983
[182] Venkatakrishnan, V., Newton solution of invisid and viscous problems, Aiaa j., 27, (1989)
[183] Venkatakrishnan, V., Viscous computations using a direct solver, Comp. fluids, 18, (1990) · Zbl 0692.76026
[184] Venkatakrishnan, V.; Mavriplis, D.J., Agglomeration multigrid for the three-dimensional Euler equations, Aiaa j., 33, 633, (1995) · Zbl 0925.76477
[185] Vesey, R.A.; Steiner, D., A 2-dimensional finite-element model of the edge plasma, J. comput. phys., 116, 300-313, (1995) · Zbl 0824.76050
[186] H.X. Vu, Plasma collection by an obstacle, PhD thesis, California Institute of Technology, 1990
[187] Wang, G.; Tafti, D.K., Performance enhancement on microprocessors with hierarchical memory systems for solving large sparse linear systems, Int. J. supercomput. appl. high perform. comput., 13, 63-79, (1999)
[188] Wesseling, P., An introduction to multigrid methods, (1992), John Wiley & Sons Chichester · Zbl 0760.65092
[189] Winkler, K.A.; Norman, M.L.; Mihalas, D., Implicit adaptive-grid radiation hydrodynamics, () · Zbl 0646.76106
[190] Wising, F.; Knoll, D.A.; Krasheninnikov, S.I.; Rognlien, T.D.; Sigmar, D.J., Simulation of the alcator C-mod divertor with an improved neutral fluid model, Contrib. plasma phys., 36, 136, (1996)
[191] Wright, S.J.; Nocedal, J., Numerical optimization, (1999), Springer Berlin · Zbl 0930.65067
[192] Xu, X.Q.; Cohen, R.H.; Rognlien, T.D.; Myra, J., Low-to-high confinement transition simulations in divertor geometry, Phys. plasmas, 7, 1951, (2000)
[193] Zanino, R., Advanced finite element modeling of the tokamak plasma edge, J. comput. phys., 138, 881-906, (1997) · Zbl 0902.76063
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.